Talk:Symmetry group

Major mistakes in section about symmetry groups of two-dimensional objects

In the section Two dimensions one assertion reads as follows:

"D₂, which is isomorphic to the Klein four-group, is the symmetry group of a non-equilateral rectangle . . . "

But this is not true. In fact the entire section is filled with mistakes like this. The symmetry group of a non-equilateral rectangle has 8 elements, not 4. Since every element is of order 2, this must be isomorphic to the group Z/2Z ⊕ Z/2Z ⊕ Z/2Z.

The section discusses only orientation-preserving symmetries, and it does this without saying so. There is no particular reason to restrict to only orientation-preserving isometries, and even though the discussion is about 2-dimensional objects there is no reason for a reader to assume that symmetries need to be orientation-preserving.

On the other hand, orientation-preserving symmetry groups (of orientable objects) are interesting in their own right, so in my opinion they should be mentioned as long as they are clearly labeled as to what they are.

And so this needs to be fixed.Daqu (talk) 16:54, 20 June 2015 (UTC)

replacing my previous edit I think we should find a suitable figure, then. —Quondum 20:45, 20 June 2015 (UTC)

Fixed: non-equilateral rectangle → non-equilateral isosceles triangle. —Quondum 20:49, 20 June 2015 (UTC)

I count only two symmetries for a non-equilateral isosceles triangle: the identity and the median through the different side. Which are the other two? --DonBex (talk) 00:02, 8 January 2016 (UTC)

This fix is wrong, the symmetry group of a non-equilateral isosceles triangle is not isomorphic to the Klein-four group. — Preceding unsigned comment added by 201.50.93.196 (talk) 22:49, 7 January 2016 (UTC)

I will start with two general statements before answering the specific questions. First, it is important to remember that this section deals with two dimensions. Some of the above statements are valid for three dimensions but not with two. And second, the elements of a group are symmetry operations such as rotations and reflections, rather than symmetry elements such as axes and planes.

In two dimensions, the isosceles triangle has only two symmetry operations = the identity operation and a single reflection in the plane which bisects the triangle. The rectangle has four symmetry operations: the identity operation, reflections in two planes, and rotation by 180^o. So the rectangle is isomorphic to the Klein-four group and the isosceles triangle is not. DonBex's original fix was therefore correct in two dimensions and I will restore it. I will also add the list of the symmetry operations.

Note however that in three dimensions, the isosceles triangle has four symmetry operations and the rectangle has eight. In the Schoenflies notation of spectroscopy and chemistry, the isosceles triangle has symmetry group C_2v which is isomorphic to the Klein-four group, while the rectangle has symmetry group D_2h. These facts may be the source of the confusion, but they are not strictly relevant to the section on two-dimensional objects. Dirac66 (talk) 01:15, 8 January 2016 (UTC)

Transformation

Shouldn't the transformations in the symmetry group of for instance a triangle be isometries?Madyno (talk) 13:35, 23 July 2017 (UTC)

I would say it depends on what one means by "triangle". If it is a geometrical triangle, then yes, the symmetry group in mind is likely the isometry. If it is a triangle graph (for example), then no, it would not be because there is no distance structure to preserve. In general, what one means by "symmetry" is always in the context of what structure one is preserving. That may include the distance, or it might not. Sławomir Biały (talk) 16:42, 23 July 2017 (UTC)

Right, yes, but my proble is that not every transformation of a triangle, that maps the triangle on itself is a symmetry. Yet the definition says so. Madyno (talk) 18:42, 23 July 2017 (UTC)

Not knowing the specific context you're referring to, I would guess that by "transformation" is intended Euclidean transformation. In that case, the statement is true, but could perhaps be clarified. Sławomir Biały (talk) 19:23, 23 July 2017 (UTC)

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"Blue arrows"?

The arrows in the figure are, roughly speaking, some sorts of pink and burnt orange. I do not edit pages any more. Perhaps someone could get cracking on this legend. 207.229.130.147 (talk) 02:39, 12 June 2020 (UTC)

Clarify that the 12 "rotations" of the tetrahedron includes the identity rotation

Re. "A tetrahedron is invariant under 12 distinct rotations, reflections excluded. These are illustrated here in the cycle graph format,": The graph shows only 11 rotations. The 12th is the "identity rotation", which is not actually a rotation; it is the rotation of no rotation. So (A) the caption should clarify that, and (B) why do we even count the identity rotation as a rotation? Philgoetz (talk) 15:30, 13 October 2021 (UTC)

I reworded the caption. I think this is much more accurate. The identity rotation is counted as a rotation to complete the rotation subgroup.—Anita5192 (talk) 16:13, 13 October 2021 (UTC)